Explore Derivatives Visually with Desmos

Build interactive Desmos graphs to visualize derivatives and verify rules.

Introduction

30 Second Summary

Every time you check how fast your car is going or watch a stock price spike, you are looking at a rate of change. Calculus calls this a derivative, and it is one of the most powerful ideas in mathematics.

In this project, you will build interactive graphs in Desmos that bring derivatives to life visually. Instead of memorizing formulas from a textbook, you will watch tangent lines glide along curves and verify every derivative rule by seeing it in action.

What You'll Build

A collection of interactive Desmos graphs that demonstrate derivative concepts through hands-on visual exploration.

By the end of this project, you'll have:

A Tangent Line Explorer graph showing secant lines converging to the tangent line as h approaches 0, proving the limit definition of the derivative.
A Derivative Rules Verifier graph confirming the power rule, product rule, quotient rule, and chain rule by graphing both sides and watching them overlap.
A Common Functions Gallery showing derivatives of polynomials, exponentials, logarithms, and trigonometric functions side-by-side.
Secret Mission: Build a derivative matching game where you graph mystery derivative curves and challenge yourself to identify the original function.

Are there any prerequisites?

You only need a basic understanding of functions (what f(x) means) and basic algebra (exponents and fractions). No prior calculus knowledge is required.

Before We Start

Before We Start...

Set Up Desmos

Before you can explore derivatives visually, you need a graphing tool that responds instantly as you experiment.

Desmos is a free, web-based graphing calculator that plots expressions the moment you type them. In this step, you will open Desmos and learn the three core interactions you will use throughout this project: typing functions, creating sliders, and plotting movable points.

In this step, get ready to:

Open the Desmos Graphing Calculator in your web browser.
Practice graphing a function and evaluating it at a specific point.
Create a slider and a movable point that follows a curve.

Open Desmos Graphing Calculator

Desmos runs entirely in your browser. No downloads, no accounts, no installation needed.

Open your web browser (Chrome, Firefox, Safari, or Edge all work).
Navigate to the Desmos Graphing Calculator by entering this URL in your address bar:

https://www.desmos.com/calculator

You should see an empty graph grid on the right and a panel called the expression list on the left.

What is the expression list?

The expression list is the left-hand panel in Desmos where you type all of your math. Each numbered row is called an expression line. You type one expression per line, and Desmos graphs it immediately on the right.

Save your work with a free account

Creating a free Desmos account lets you save and revisit your graphs later. Click the hamburger menu (three horizontal lines) in the top-left corner, then select Sign In to create an account or sign in with Google. This is optional for this project.

Type your first function

Every derivative concept in this project starts with a function. Let's confirm you can define one and evaluate it.

Click on the first expression line in the expression list (the blank row labeled 1 on the left).
Type the following to define a function:

f(x) = x^2

You should see a parabola (U-shaped curve) appear on the graph immediately.

Click on the next empty expression line (row 2).
Type the following to evaluate your function at a specific point:

f(3)

Desmos should display the result 9 next to the expression. This confirms that f(3) = 3 squared = 9.

What does this do?

When you type f(x) = x^2, you are defining a function named f that squares any input value x. Desmos graphs it instantly so you can see its shape.

When you type f(3) on a new line, Desmos evaluates the function at x = 3 and shows you the numeric result. You will use this technique later to check derivative values at specific points.

Don't see the parabola?

Make sure you typed f(x) = x^2 with the parentheses around x and the caret (^) symbol for the exponent. If you see a red error message on the expression line, delete the text and retype it carefully.

help me fix my Desmos expression that is showing an error

Discover sliders and movable points

Sliders let you create a variable you can drag to change in real time. Combined with a function, they let you trace a point along a curve. This is the foundation for visualizing tangent lines in the next step.

Click on the next empty expression line (row 3).
Type the following to create a slider variable:

a = 1

Desmos creates a draggable slider for the variable a with a default range of -10 to 10. You can drag the slider handle left and right to change the value of a.

Click on the next empty expression line (row 4).
Type the following to plot a point on your parabola that moves with the slider:

(a, f(a))

Drag the slider for a left and right. Watch the point glide along the parabola.

What is happening here?

The expression (a, f(a)) creates a coordinate point where the x-value is whatever a currently equals, and the y-value is f evaluated at that same a. Since a is a slider, the point updates in real time as you drag.

In the next step, you will attach a tangent line to this movable point. The slope of that tangent line is the derivative at that point.

No slider appeared?

If you typed a = 1 but no slider appeared, make sure you are typing on a fresh empty expression line. Also confirm you typed a lowercase a, not a capital A. Desmos treats uppercase and lowercase letters differently.

help me get my Desmos slider working

✔️ Awesome, I've got everything!

Great. Make sure you can see the parabola, the value 9 next to f(3), and a point that moves along the curve when you drag the slider.

ⓧ I'd like to double check the full expressions

Your expression list should contain these four lines in order:

f(x) = x^2
f(3)
a = 1
(a, f(a))

Your Desmos workspace is ready. Next up, you will use these exact skills to build a secant line that transforms into a tangent line, showing you the derivative definition in action.

Discover What a Derivative Is

In the previous step, you got comfortable with Desmos by graphing a function, creating a slider, and plotting a point.

Now you will build the core visual that makes derivatives click. You will watch a secant line rotate into a tangent line as a gap shrinks to zero. This is the limit definition of the derivative happening right in front of you.

In this step, get ready to:

Build a secant line that converges into a tangent line as h approaches 0.
Add Desmos's exact tangent line and confirm it matches your secant line at the limit.
Visualize instantaneous rate of change by sliding a tangent point along a curve.

Build a secant line that becomes a tangent

A secant line passes through two points on a curve.

As you bring those two points closer together, the secant line rotates until it touches the curve at exactly one point. That final position is the tangent line, and its slope is the derivative.

Start a fresh graph by opening a new browser tab and navigating to https://www.desmos.com/calculator.
Define the function by typing the following on the first expression line:

f(x) = x^2

Create a slider for the point of interest by typing on the next empty expression line:

a = 1

Create a slider for the interval size by typing on the next empty expression line:

h = 1

Type the secant line formula on the next expression line:

y = ((f(a+h) - f(a))/h)(x - a) + f(a)

What does this expression do?

This is the point-slope form of a line through the point (a, f(a)) with a slope equal to the difference quotient (f(a+h) - f(a))/h.

The difference quotient measures the average rate of change of the function between two points separated by distance h. As h shrinks to 0, that average becomes an instantaneous rate of change.

Drag the h slider slowly toward 0.
Watch the secant line rotate as the two points on the curve get closer together.

When h is very close to 0, the secant line barely moves anymore. It has settled into the tangent line position.

Line not changing as you drag h?

Make sure the slider for h is the one you are dragging, not a. The a slider moves the point along the curve, while h controls the gap between the two secant points.

Also check that your secant line expression matches y = ((f(a+h) - f(a))/h)(x - a) + f(a) exactly. Missing parentheses will break the slope calculation.

Help me troubleshoot my secant line

Add the exact tangent line for comparison

Desmos can compute derivatives directly using prime notation. You will now add the exact tangent line so you can see it overlap with your secant line as h gets small.

Type the exact tangent line on the next empty expression line:

y = f'(a)(x - a) + f(a)

What is f'(a)?

The notation f'(a) tells Desmos to compute the derivative of f at the point a. This gives the exact slope of the tangent line at that point.

The full expression y = f'(a)(x - a) + f(a) is the tangent line in point-slope form. It passes through (a, f(a)) with slope f'(a).

Drag the h slider toward 0 again.
Watch your secant line converge onto the exact tangent line until both lines overlap completely.

This overlap is visual proof that the limit definition produces the same result as the derivative.

Why two lines?

The secant line uses the difference quotient formula, which is an approximation when h is not zero. The tangent line uses the exact derivative.

Watching them merge as h approaches 0 proves these two approaches give the same slope at the limit.

Lines not overlapping?

Set h to a very small value like 0.01 or 0.001. If the lines still do not overlap, check that both expressions reference the same function f(x) = x^2 defined on line 1.

Help me fix my tangent and secant lines

Visualize rate of change

The derivative tells you how fast a function is changing at any given point. You will now see this in action on a wave-shaped curve where the rate of change varies dramatically.

Click on line 1 (where f(x) = x^2 is) and replace it with the following:

f(x) = sin(x) + 3

Why replace the function?

Since your tangent line expression y = f'(a)(x - a) + f(a) references f(x), changing line 1 automatically updates the tangent line to work with the new curve. No need to retype anything else.

Drag the a slider slowly from left to right across the graph.
Watch how the tangent line tilts as the point moves along the sine curve.

What are you seeing?

Where the sine curve rises steeply, the tangent line tilts upward. A positive slope means the function is increasing quickly.

Where the curve reaches a peak, the tangent line goes perfectly flat. Zero slope means no change at that instant.

Where the curve falls, the tangent line tilts downward. Negative slope means the function is decreasing. This is instantaneous rate of change in action.

Try clicking the play button next to the a slider to animate the tangent point gliding along the curve automatically.

Use animation to spot patterns

The play button next to every slider animates through all values. This makes it easy to see the tangent line steepen, flatten, and steepen again in a repeating cycle that matches the sine wave's rhythm.

Tangent line not moving with the slider?

Confirm that your tangent line expression uses the variable a (lowercase), matching the slider name exactly. If you accidentally typed a capital A or a different letter, the line will not respond to the slider.

Help me fix my tangent line animation

✔️ Awesome, I've got everything!

Great. Make sure all five expressions are in your graph before moving on.

ⓧ I'd like to double check the full expressions

Here are all the expressions you should have in your Desmos graph at this point:

f(x) = sin(x) + 3
a = 1
h = 1
y = ((f(a+h) - f(a))/h)(x - a) + f(a)
y = f'(a)(x - a) + f(a)

Line 1 was originally f(x) = x^2 and you changed it to f(x) = sin(x) + 3 in the final substep. You can change line 1 back to f(x) = x^2 any time to revisit the secant-to-tangent convergence.

You have just seen the fundamental idea behind derivatives: the slope of the tangent line at any point IS the derivative at that point. Next up, you will put this knowledge to work by verifying the basic derivative rules visually.

Verify Basic Derivative Rules

In the last step, you watched a secant line rotate into a tangent line, proving the limit definition of the derivative with your own eyes. Now you know what a derivative is. The next question is: how do you compute one quickly?

Mathematicians have developed a set of rules that let you find derivatives without going through the limit definition every time. In this step, you will verify these rules visually by graphing both sides and confirming they produce identical curves.

In this step, get ready to:

Verify the constant rule and power rule.
Verify the sum rule and difference rule.
Verify the constant multiple rule.

Verify the constant rule and power rule

The constant rule says the derivative of any constant is zero. The power rule says the derivative of x^n is n*x^(n-1). You will confirm both by graphing the derivatives and checking they match your hand-computed predictions.

Open a fresh Desmos graph by navigating to https://www.desmos.com/calculator in a new browser tab.
Type the following expressions into the expression list, one per line:

f(x) = 5
f'(x)
g(x) = x^3
g'(x)
3x^2

What do these expressions do?

f(x) = 5 defines a constant function (a flat horizontal line at y = 5).
f'(x) graphs the derivative of f using Desmos's built-in prime notation. Since f never changes, the derivative is 0 everywhere.
g(x) = x^3 defines a cubic function.
g'(x) graphs the derivative of x^3 using Desmos's automatic differentiation.
3x^2 is your hand-computed prediction using the power rule (bring down the 3, reduce the exponent by 1). If the rule is correct, this curve and g'(x) should overlap perfectly.

Look at your graph. You should see two things confirming the rules work.

Confirm that f'(x) appears as a flat line sitting on the x-axis (y = 0). This proves the constant rule.
Confirm that g'(x) and 3x^2 overlap completely. You should see only one curve because they are drawn on top of each other. This proves the power rule.

Curves not overlapping?

Make sure g(x) = x^3 uses the caret (^) for the exponent, not the letter x typed three times.

Check that 3x^2 is on its own expression line, not appended to a previous line. Each expression must be on a separate line in the expression list.

If you see a red error highlight on a line, click it and check for missing parentheses or stray characters.

Still stuck? Help me figure out why my Desmos derivative curves don't overlap.

Verify the sum and difference rules

The sum rule says you can take the derivative of each term and add them. The difference rule says the same for subtraction. You will define two functions and verify that Desmos's derivative matches your hand-computed result for each.

Add these expressions on new lines below your existing ones:

p(x) = x^3 + x^2
p'(x)
3x^2 + 2x
q(x) = x^3 - x^2
q'(x)
3x^2 - 2x

What's happening here?

For p(x) = x^3 + x^2, the sum rule says: take the derivative of each term separately. The derivative of x^3 is 3x^2, and the derivative of x^2 is 2x. Add them to get 3x^2 + 2x.

For q(x) = x^3 - x^2, the difference rule works the same way but with subtraction. The derivative is 3x^2 - 2x.

If the rules are correct, Desmos's automatically computed p'(x) will overlap perfectly with 3x^2 + 2x, and q'(x) will overlap with 3x^2 - 2x.

Confirm that p'(x) and 3x^2 + 2x overlap exactly on the graph.
Confirm that q'(x) and 3x^2 - 2x overlap exactly on the graph.

Too many curves on screen?

Click the colored circle next to any expression to toggle its visibility. This lets you isolate just the pair you want to compare without deleting anything.

Seeing a different curve shape?

Double check that p(x) uses a plus sign and q(x) uses a minus sign. A typo here changes the entire function shape.

Make sure each prediction (3x^2 + 2x and 3x^2 - 2x) is on its own expression line.

Need help? Why doesn't my sum rule verification work in Desmos?

Verify Advanced Derivative Rules

You have confirmed that the basic derivative rules (constant, power, sum, difference, constant multiple) all produce the correct slopes. Now it is time to tackle the three rules that handle combinations of functions.

When two functions are multiplied, divided, or nested inside each other, you need the product rule, quotient rule, or chain rule. In this step, you will verify all three by graphing both Desmos's built-in derivative and the rule's formula side-by-side.

In this step, get ready to:

Verify the product rule by graphing the derivative of two multiplied functions.
Verify the quotient rule by graphing the derivative of one function divided by another.
Verify the chain rule by graphing the derivative of a function nested inside another.

Verify the product rule

The product rule tells you how to differentiate two functions multiplied together. The formula is: the derivative of f(x) times g(x) equals f'(x)*g(x) + f(x)*g'(x).

Open a new Desmos graph by navigating to https://www.desmos.com/calculator in a new browser tab.
Define the two base functions by typing the following expressions into the expression list, one per line:

f(x) = x^2
g(x) = sin(x)

Define the product and graph its derivative by typing these expressions on new lines:

p(x) = f(x)*g(x)
p'(x)

Graph the product rule formula on a new line to verify it matches:

f'(x)*g(x) + f(x)*g'(x)

What does this verify?

You now have two curves on the graph: the built-in derivative p'(x) and the product rule formula f'(x)*g(x) + f(x)*g'(x). If the product rule is correct, these two curves overlap perfectly.

The rule says: to differentiate a product, take the derivative of the first times the second, plus the first times the derivative of the second. You are seeing this formula confirmed visually.

You should see both derivative curves overlapping exactly on your graph. They will trace the same path because the product rule formula produces the correct derivative.

Curves not overlapping?

Make sure you typed f(x)*g(x) with the asterisk for multiplication. Desmos does not always interpret two adjacent parentheses as multiplication.
Check that your product rule formula has both terms separated by a plus sign: f'(x)*g(x) + f(x)*g'(x).
Confirm that f(x) and g(x) are defined on earlier lines. Desmos cannot compute a derivative of an undefined function.

Why are my Desmos derivative curves not overlapping?

Verify the quotient rule

The quotient rule handles the derivative of one function divided by another. The formula is: (f'g - fg') / g^2.

Define the quotient function by typing this on a new expression line in the same graph:

q(x) = sin(x)/x

Graph the built-in derivative on the next line:

q'(x)

Graph the quotient rule formula on a new line to verify it matches:

(cos(x)*x - sin(x)*1)/x^2

How does the quotient rule formula work here?

For sin(x)/x, the "top" function is sin(x) and the "bottom" function is x. The quotient rule says: (derivative of top times bottom, minus top times derivative of bottom) all divided by bottom squared.

The derivative of sin(x) is cos(x), and the derivative of x is 1. That gives you (cos(x)*x - sin(x)*1)/x^2.

You should see both curves tracing the same path. Notice the gap at x = 0 where the function is undefined (you cannot divide by zero).

Seeing a gap or mismatch near x = 0?

A gap at x = 0 is expected. Both sin(x)/x and its derivative are undefined at zero because of division by zero.
If the curves do not overlap everywhere else, check that you typed cos(x)*x (not cos(x*x) which means something different).

Help me fix my quotient rule verification in Desmos.

Verify the chain rule

The chain rule handles a function nested inside another function. It says: the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). You differentiate the "outer" function, then multiply by the derivative of the "inner" function.

Define the composite function by typing this on a new expression line:

c(x) = sin(x^2)

Graph the built-in derivative on the next line:

c'(x)

Graph the chain rule result on a new line to verify it matches:

cos(x^2)*2x

How does the chain rule apply here?

In sin(x^2), the "outer" function is sin(...) and the "inner" function is x^2. The chain rule says: differentiate the outer function first (giving cos(x^2)), then multiply by the derivative of the inner function (giving 2x).

The final result is cos(x^2)*2x. This is the most commonly used rule in calculus because most real-world functions involve compositions.

You should see both curves overlapping exactly. The derivative oscillates more and more rapidly as x grows, because x^2 inside sin changes faster at larger values of x.

Curves look different or only partially overlap?

Make sure you typed cos(x^2)*2x and not cos(2x)*2x. The argument inside cosine must be x^2 (the same inner function).
Zoom out using the minus button on the graph if the curves look like they diverge. At higher zoom levels both curves trace the same oscillating pattern.

Help me troubleshoot my chain rule graph in Desmos.

Why do overlapping curves prove the rule?

Desmos computes p'(x), q'(x), and c'(x) using the formal limit definition of the derivative. When your hand-computed formula produces the exact same curve, it proves the rule gives the correct answer for all values of x simultaneously.

If you would like to save this graph, click the dropdown next to New Graph in the upper left. You can name it something like "Derivative Rules Verifier (Advanced)" to find it later.

✔️ Awesome, I've got everything!

Great. Make sure you can see both curves overlapping for each of the three rules before moving on.

ⓧ I'd like to double check the full expressions

Here are all the expressions that should be in your advanced derivative rules graph:

f(x) = x^2
g(x) = sin(x)
p(x) = f(x)*g(x)
p'(x)
f'(x)*g(x) + f(x)*g'(x)
q(x) = sin(x)/x
q'(x)
(cos(x)*x - sin(x)*1)/x^2
c(x) = sin(x^2)
c'(x)
cos(x^2)*2x

You have now confirmed the three most powerful derivative rules with your own eyes. Next up, you will use these rules to explore the derivatives of common function families: polynomials, exponentials, logarithms, and trigonometric functions.

Explore Derivatives of Common Functions

You have now confirmed that every major derivative rule produces the correct result. The rules work. The next question is: what do the derivatives of real, everyday functions actually look like?

In this step, you will build a visual reference library of common function derivatives. You will graph polynomials, exponentials, logarithms, and trig functions alongside their derivatives to see the patterns that make calculus click.

In this step, get ready to:

Graph a polynomial and its derivative, then trace slope changes with a tangent line.
Discover that e^x is its own derivative and that ln(x) differentiates to 1/x.
Verify the derivatives of sin(x), cos(x), and tan(x) visually.

Graph a polynomial and trace its slope

Polynomials are the most common functions in calculus. Their derivatives follow a simple pattern: the power rule drops the exponent by one on every term.

Start a new graph by clicking the hamburger menu (three horizontal lines) in the top-left corner of Desmos and selecting New Blank Graph.
Type the following expressions, each on its own line in the expression list:

f(x) = 2x^4 - 3x^2 + x - 7
f'(x)

What do these expressions do?

The first line defines a degree-4 polynomial with multiple terms. The second line uses Desmos's prime notation to automatically graph the derivative.

Notice that the derivative curve is one degree lower than the original. A degree-4 polynomial always produces a degree-3 derivative. This is the power rule in action across every term.

Now add a tangent line with a movable point so you can trace how the slope changes across the curve. Type these on new lines:

a = 1
y = f'(a)(x - a) + f(a)

What does the tangent line show?

The slider a controls which point on the curve the tangent line touches. The formula y = f'(a)(x - a) + f(a) builds a line with slope equal to the derivative at that point, passing through the point (a, f(a)).

As you drag the slider, the tangent line tilts to match the derivative value at every location. Where the derivative graph crosses zero, the tangent line goes flat.

Drag the a slider left and right. Watch how the tangent line's steepness matches the height of the derivative curve at the same x-value.

You should see two curves on the graph (the polynomial and its derivative) plus a straight tangent line that rotates as you move the slider.

Don't see the tangent line?

Make sure the slider line reads exactly a = 1 with no extra characters. If Desmos shows a red error indicator on the tangent line, check that you typed f'(a) (with the apostrophe) and not f(a) alone.

Why isn't my tangent line showing up in Desmos?

Explore exponential and logarithmic derivatives

Exponential and logarithmic functions have some of the most elegant derivatives in all of calculus. You are about to see why mathematicians consider e so special.

You will reuse the function name f(x) for this section. Hide the polynomial expressions by clicking the colored circle next to each of the previous lines (the polynomial, its derivative, the slider, and the tangent line). This toggles their visibility without deleting them.
Type the following on new expression lines:

f(x) = e^x
f'(x)

What is happening here?

Look closely at the graph. The derivative curve sits directly on top of the original e^x curve. They are the same function. This is the remarkable property of e^x: it is its own derivative. The rate of change of e^x at any point equals its own value at that point.

Verify the overlap by clicking on either curve. If both curves highlight together, they are identical.
Now add the natural logarithm and its derivative on new lines:

g(x) = ln(x)
g'(x)
1/x

What should I see?

The g'(x) curve and the 1/x curve overlap perfectly. This confirms that the derivative of ln(x) is 1/x. Notice how the derivative is large near x = 0 (the logarithm is rising steeply there) and approaches zero as x grows (the logarithm flattens out).

You should see the exponential curves overlapping on one part of the graph and the logarithm curves overlapping on another.

Seeing a 'definition conflict' warning?

If Desmos shows a yellow triangle on your f(x) lines, it means two definitions of f(x) are both visible. Click the colored circle next to the polynomial f(x) line from the previous section to hide it. Only one definition of the same function name can be active at a time.

Help me fix the definition conflict in my Desmos graph

Verify trigonometric function derivatives

Trigonometric derivatives follow a cycle: sin becomes cos, cos becomes negative sin, and the cycle repeats. Seeing this visually makes the pattern impossible to forget.

Hide the exponential and logarithmic expressions by clicking their colored circles.
Type the following expressions for sin(x) on new lines:

f(x) = sin(x)
f'(x)
cos(x)

Look at the graph. The f'(x) curve and the cos(x) curve should overlap exactly, confirming the derivative of sin(x) is cos(x).
Now hide those three lines and type the cosine expressions on new lines:

g(x) = cos(x)
g'(x)
-sin(x)

Verify that g'(x) overlaps with -sin(x). The derivative of cos(x) is negative sin(x). Notice the negative sign flips the sine wave upside down.
Finally, type the tangent expressions on new lines:

h(x) = tan(x)
h'(x)
1/cos(x)^2

What does the tangent derivative look like?

The derivative of tan(x) is 1/cos(x)^2, also written as sec^2(x). The h'(x) curve and the 1/cos(x)^2 curve overlap exactly, confirming this rule.

Notice that the derivative is always positive (or undefined at the asymptotes). This makes sense because tan(x) is always increasing between its vertical asymptotes.

You should see the derivative curve matching the verification curve on each pair of trig expressions.

Trig curves not overlapping?

Make sure you typed -sin(x) with the negative sign for the cosine derivative. If you typed sin(x) without the negative, the curves will be mirror images instead of overlapping.

For tan(x), check that you typed 1/cos(x)^2 and not 1/cos(x^2). The exponent applies to cos(x), not to x itself.

Why aren't my trig derivative curves overlapping in Desmos?

✔️ Awesome, I've got everything!

Great. Make sure all your expressions are saved in this graph. You now have a complete visual reference library.

ⓧ I'd like to double check the full expression list

Here is the full list of expressions that should exist in your Common Functions Gallery graph. Some will be hidden (toggled off) to avoid definition conflicts, but all should be present in the expression list:

# Polynomial section (may be hidden)
f(x) = 2x^4 - 3x^2 + x - 7
f'(x)
a = 1
y = f'(a)(x - a) + f(a)

# Exponential and logarithmic section (may be hidden)
f(x) = e^x
f'(x)
g(x) = ln(x)
g'(x)
1/x

# Trigonometric section
f(x) = sin(x)
f'(x)
cos(x)
g(x) = cos(x)
g'(x)
-sin(x)
h(x) = tan(x)
h'(x)
1/cos(x)^2

You now have a complete visual reference library for the most important function families in calculus. Next up, you will put all of this derivative knowledge to the test in a creative challenge.

Secret mission

Build a Derivative Matching Game

Can you look at a derivative curve and figure out what the original function looks like? Build an interactive matching game in Desmos that tests your derivative intuition by graphing mystery derivatives and challenging you to reason backward to the original functions.

Clean Up Your Resources

This project runs entirely in the browser using Desmos, so there are no ongoing costs or cloud resources to manage.

Decide whether to keep your Desmos graphs accessible, save them for later, or remove them entirely.

Resources you used:

Desmos graph: Tangent Line Explorer.
Desmos graph: Derivative Rules Verifier (Basic).
Desmos graph: Derivative Rules Verifier (Advanced).
Desmos graph: Common Functions Gallery.
Desmos graph: Derivative Matching Game.

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Save your work to a free Desmos account so you can close your browser tabs and return later.

If you have not created a Desmos account yet, click the sign-in icon in the top-right corner of the Desmos calculator and create a free account.
For each graph tab, click the hamburger menu (three lines) in the top-left corner, then click Save. Give each graph a descriptive name (e.g. "Tangent Line Explorer", "Basic Rules Verifier").
Once all five graphs are saved, you can safely close the browser tabs. Reopen them anytime from your Desmos account dashboard.

Delete - I don't want to use this again

Remove all project graphs and start fresh.

If your graphs are NOT saved to an account, simply close all five Desmos browser tabs. The graphs are gone permanently once the tabs are closed.
If your graphs ARE saved to a Desmos account, open your saved graphs list by clicking the hamburger menu (three lines) in the top-left corner, then click Open Graph. Find each graph (Tangent Line Explorer, Derivative Rules Verifier Basic, Derivative Rules Verifier Advanced, Common Functions Gallery, Derivative Matching Game), click the three-dot menu next to it, and select Move to Trash.
Repeat for all five graphs to remove them from your account.

Nice Work!

Nice work! You have built a complete visual understanding of derivatives from the ground up using interactive Desmos graphs.

You've learned how to:

Built a tangent line explorer showing the limit definition of the derivative in action, watching secant lines converge to tangent lines as h approaches 0.
Verified the constant rule, power rule, sum rule, difference rule, product rule, quotient rule, and chain rule by graphing both sides of each rule and confirming the curves overlap exactly.
Explored derivatives of polynomials, exponentials, logarithms, and trigonometric functions visually, building a reference gallery of common derivatives and their behaviors.
Built a derivative matching game that tests your ability to reason backward from a derivative curve to identify the original function.

Ready to quiz yourself?